Séminaire du LPTMS: Baruch Meerson

Geometrical optics of constrained Brownian motion: three short stories

Baruch Meerson (Racah Institute of Physics, Hebrew University of Jerusalem)

The optimal fluctuation method — essentially geometrical optics — gives a deep insight into large deviations of Brownian motion, and it achieves this purpose by simple means. Here we illustrate these points by telling three short stories about Brownian motions, « pushed » into a large-deviation regime by constraints.

Story 1 deals with a long-time survival of a Brownian particle in 1 + 1 dimension against absorption by a wall which advances according to a power law xw(t)~ tγ, where γ > 1/2. We also calculate the large deviation function (LDF) of the particle position at an earlier time, conditional on the survival by a later time.

Story 2 addresses a stretched Brownian motion above an absorbing obstacle in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point.

In story 3 we compute the short-time LDF of the winding angle of a Brownian particle wandering around a reflecting disk in the plane.

In all three stories we uncover singularities of the LDFs which can be interpreted as dynamical phase transitions and which have a simple geometric origin. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations. We argue that, in stories 1 and 2, this is the Ferrari–Spohn distribution.

The talk is based on a recent paper by B. Meerson and N. R. Smith, J. Phys. A: Math. Theor. 52, 415001 (2019).


Date/Time : 21/01/2020 - 11:00 - 12:00

Location : Salle des séminaires du FAST et du LPTMS, bâtiment Pascal n°530

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